Mathbox for Richard Penner < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dfsucon Structured version   Visualization version   GIF version

Theorem dfsucon 40161
 Description: 𝐴 is called a successor ordinal if it is not a limit ordinal and not the empty set. (Contributed by RP, 11-Nov-2023.)
Assertion
Ref Expression
dfsucon ((Ord 𝐴 ∧ ¬ Lim 𝐴𝐴 ≠ ∅) ↔ ∃𝑥 ∈ On 𝐴 = suc 𝑥)
Distinct variable group:   𝑥,𝐴

Proof of Theorem dfsucon
StepHypRef Expression
1 3ancomb 1096 . . . 4 ((Ord 𝐴 ∧ ¬ Lim 𝐴𝐴 ≠ ∅) ↔ (Ord 𝐴𝐴 ≠ ∅ ∧ ¬ Lim 𝐴))
2 df-3an 1086 . . . 4 ((Ord 𝐴𝐴 ≠ ∅ ∧ ¬ Lim 𝐴) ↔ ((Ord 𝐴𝐴 ≠ ∅) ∧ ¬ Lim 𝐴))
3 df-ne 3012 . . . . . . . . 9 (𝐴 ≠ ∅ ↔ ¬ 𝐴 = ∅)
43anbi2i 625 . . . . . . . 8 ((Ord 𝐴𝐴 ≠ ∅) ↔ (Ord 𝐴 ∧ ¬ 𝐴 = ∅))
54imbi1i 353 . . . . . . 7 (((Ord 𝐴𝐴 ≠ ∅) → ∃𝑥 ∈ On 𝐴 = suc 𝑥) ↔ ((Ord 𝐴 ∧ ¬ 𝐴 = ∅) → ∃𝑥 ∈ On 𝐴 = suc 𝑥))
6 pm5.6 999 . . . . . . 7 (((Ord 𝐴 ∧ ¬ 𝐴 = ∅) → ∃𝑥 ∈ On 𝐴 = suc 𝑥) ↔ (Ord 𝐴 → (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥)))
7 iman 405 . . . . . . 7 ((Ord 𝐴 → (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥)) ↔ ¬ (Ord 𝐴 ∧ ¬ (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥)))
85, 6, 73bitrri 301 . . . . . 6 (¬ (Ord 𝐴 ∧ ¬ (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥)) ↔ ((Ord 𝐴𝐴 ≠ ∅) → ∃𝑥 ∈ On 𝐴 = suc 𝑥))
9 dflim3 7547 . . . . . 6 (Lim 𝐴 ↔ (Ord 𝐴 ∧ ¬ (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥)))
108, 9xchnxbir 336 . . . . 5 (¬ Lim 𝐴 ↔ ((Ord 𝐴𝐴 ≠ ∅) → ∃𝑥 ∈ On 𝐴 = suc 𝑥))
1110anbi2i 625 . . . 4 (((Ord 𝐴𝐴 ≠ ∅) ∧ ¬ Lim 𝐴) ↔ ((Ord 𝐴𝐴 ≠ ∅) ∧ ((Ord 𝐴𝐴 ≠ ∅) → ∃𝑥 ∈ On 𝐴 = suc 𝑥)))
121, 2, 113bitri 300 . . 3 ((Ord 𝐴 ∧ ¬ Lim 𝐴𝐴 ≠ ∅) ↔ ((Ord 𝐴𝐴 ≠ ∅) ∧ ((Ord 𝐴𝐴 ≠ ∅) → ∃𝑥 ∈ On 𝐴 = suc 𝑥)))
13 pm3.35 802 . . 3 (((Ord 𝐴𝐴 ≠ ∅) ∧ ((Ord 𝐴𝐴 ≠ ∅) → ∃𝑥 ∈ On 𝐴 = suc 𝑥)) → ∃𝑥 ∈ On 𝐴 = suc 𝑥)
1412, 13sylbi 220 . 2 ((Ord 𝐴 ∧ ¬ Lim 𝐴𝐴 ≠ ∅) → ∃𝑥 ∈ On 𝐴 = suc 𝑥)
15 eloni 6179 . . . . . 6 (𝑥 ∈ On → Ord 𝑥)
16 ordsuc 7514 . . . . . 6 (Ord 𝑥 ↔ Ord suc 𝑥)
1715, 16sylib 221 . . . . 5 (𝑥 ∈ On → Ord suc 𝑥)
18 nlimsucg 7542 . . . . 5 (𝑥 ∈ On → ¬ Lim suc 𝑥)
19 nsuceq0 6249 . . . . . 6 suc 𝑥 ≠ ∅
2019a1i 11 . . . . 5 (𝑥 ∈ On → suc 𝑥 ≠ ∅)
2117, 18, 203jca 1125 . . . 4 (𝑥 ∈ On → (Ord suc 𝑥 ∧ ¬ Lim suc 𝑥 ∧ suc 𝑥 ≠ ∅))
22 ordeq 6176 . . . . 5 (𝐴 = suc 𝑥 → (Ord 𝐴 ↔ Ord suc 𝑥))
23 limeq 6181 . . . . . 6 (𝐴 = suc 𝑥 → (Lim 𝐴 ↔ Lim suc 𝑥))
2423notbid 321 . . . . 5 (𝐴 = suc 𝑥 → (¬ Lim 𝐴 ↔ ¬ Lim suc 𝑥))
25 neeq1 3073 . . . . 5 (𝐴 = suc 𝑥 → (𝐴 ≠ ∅ ↔ suc 𝑥 ≠ ∅))
2622, 24, 253anbi123d 1433 . . . 4 (𝐴 = suc 𝑥 → ((Ord 𝐴 ∧ ¬ Lim 𝐴𝐴 ≠ ∅) ↔ (Ord suc 𝑥 ∧ ¬ Lim suc 𝑥 ∧ suc 𝑥 ≠ ∅)))
2721, 26syl5ibrcom 250 . . 3 (𝑥 ∈ On → (𝐴 = suc 𝑥 → (Ord 𝐴 ∧ ¬ Lim 𝐴𝐴 ≠ ∅)))
2827rexlimiv 3266 . 2 (∃𝑥 ∈ On 𝐴 = suc 𝑥 → (Ord 𝐴 ∧ ¬ Lim 𝐴𝐴 ≠ ∅))
2914, 28impbii 212 1 ((Ord 𝐴 ∧ ¬ Lim 𝐴𝐴 ≠ ∅) ↔ ∃𝑥 ∈ On 𝐴 = suc 𝑥)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 844   ∧ w3a 1084   = wceq 1538   ∈ wcel 2114   ≠ wne 3011  ∃wrex 3131  ∅c0 4265  Ord word 6168  Oncon0 6169  Lim wlim 6170  suc csuc 6171 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pr 5307  ax-un 7446 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-rab 3139  df-v 3471  df-sbc 3748  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-tp 4544  df-op 4546  df-uni 4814  df-br 5043  df-opab 5105  df-tr 5149  df-eprel 5442  df-po 5451  df-so 5452  df-fr 5491  df-we 5493  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator